simon-5502-08-slides

Topics to be covered

What you will learn
- A simple example of survival data
- Overall Kaplan-Meier curve
- The log rank test
- The hazard function
- The Cox regression model
- Assumptions and data management

Survival analysis

Time to event models
- Death
- Relapse
- Rehospitalization
- Failure of medical device
- Pregnancy
Not every patient experiences the event
- These are censored observations

Survival analysis models are more properly called time to event models. You follow a group of patients from a certain time point and note the amount of time until they die.

Or the amount of time until they relapse. Or the amount of time until they need to be rehospitalized. Or the amount time until a medical device that you implanted in them fails.

I should note that while almost all of the events in a time to event model are bad, there are a few exceptions. In a study of couples with fertility problems, you might use a time to event model to study the time to pregnancy, a very happy event for any couple with fertility problems.

Mortality is the context under which time to event models were derived, so the term survival analysis has been used even when the event is different.

A key feature of survival analysis is that not every patient experiences the event. You should be glad that not everyone that you recruit for a clinical trial dies, but this adds a layer of complexity to the analysis.

The reasons for not experiencing the event can vary. Everyone dies, but maybe the event is death from cancer and if you patient gets hit by a bus, that patient does not experience the event. If the event is death from any cause, you still have to end the study in some time frame, and not everyone will die in that time frame. Not every patient gets rehospitalized, not every patient relapses, not every medical device fails.

A patient may drop out of a study, and you no longer are able to tell from that point onward whether that patient would have experienced the event sometime during the rest of your study.

When your patient does not die during the study, this is not a missing value. You have partial information. You know that the patient was alive for a certain amount of time. When you end the study within a certain time frame, you know that your patient dies at a time beyond the end date of your study. If your patient drops out after six months, you know that the patient survived for more than six months.

First fruit fly experiment, 1

data_dictionary: fly1.txt

description: |
  This dataset provides a simple example of what survival and censoring. It provides an inuitive explanation of estimation of survival probabilities.

vars:
  day:
    label: Time until death
    unit: days

First fruit fly experiment, 2

37, 40, 43, 44, 45, 47, 49, 54, 56, 58, 59, 60, 61, 62, 68, 70, 71, 72, 73, 75, 77, 79, 89, 94, 96

First fruit fly experiment, 3

First fruit fly experiment, 4

Second fruit fly experiment, 1

37, 40, 43, 44, 45, 47, 49, 54, 56, 58, 59, 60, 61, 62, 68, ??, ??, ??, ??, ??, ??, ??, ??, ??, ??

Second fruit fly experiment, 2

  day event
1  37     1
2  40     1
3  43     1
4  44     1
5  45     1
6  47     1
7  49     1
8  54     1
9  56     1

   day event
10  58     1
11  59     1
12  60     1
13  61     1
14  62     1
15  68     1
16  70     0
17  70     0
18  70     0

   day event
19  70     0
20  70     0
21  70     0
22  70     0
23  70     0
24  70     0
25  70     0

Second fruit fly experiment, 3

  day event   p
1  37     1 96%
2  40     1 92%
3  43     1 88%
4  44     1 84%
5  45     1 80%
6  47     1 76%
7  49     1 72%
8  54     1 68%
9  56     1 64%

   day event   p
10  58     1 60%
11  59     1 56%
12  60     1 52%
13  61     1 48%
14  62     1 44%
15  68     1 40%
16  70     0    
17  70     0    
18  70     0

   day event p
19  70     0  
20  70     0  
21  70     0  
22  70     0  
23  70     0  
24  70     0  
25  70     0

Second fruit fly experiment, 4

Third fruit fly experiment, 1

37, 40, 43, 44, 45, 47, 49, 54, 56, 58, 59, 60, 61, 62, 68, ??, 71, ??, ??, 75, ??, ??, 89, ??, 96

Third fruit fly experiment, 2

  day event
1  37     1
2  40     1
3  43     1
4  44     1
5  45     1
6  47     1
7  49     1
8  54     1
9  56     1

   day event
10  58     1
11  59     1
12  60     1
13  61     1
14  62     1
15  68     1
16  70     0
17  71     1
18  70     0

   day event
19  70     0
20  75     1
21  70     0
22  70     0
23  89     1
24  70     0
25  96     1

Third fruit fly experiment, 3

  day event   p
1  37     1 96%
2  40     1 92%
3  43     1 88%
4  44     1 84%
5  45     1 80%
6  47     1 76%
7  49     1 72%
8  54     1 68%
9  56     1 64%

   day event   p
10  58     1 60%
11  59     1 56%
12  60     1 52%
13  61     1 48%
14  62     1 44%
15  68     1 40%
16  70     0    
17  71     1 30%
18  70     0

   day event   p
19  70     0    
20  75     1 20%
21  70     0    
22  70     0    
23  89     1 10%
24  70     0    
25  96     1  0%

Third fruit fly experiment, 4

Here is a graph of the survival probability estimates from the third experiment.

These survival probabilities differ only slightly from the survival probabilities in the original experiment. This works out because the mechanism that caused us to lose information on six of the fruit flies was independent of their ultimate survival.

If the censoring mechanism were somehow related to survival prognosis, then you would have the possibility of serious bias in your estimates. Suppose for example, that only the toughest of flies (those with the most days left in their short lives) would have been able to escape. The flies destined to die on days 70, 71, 72, and 73, were already on their deathbeds and unable to fly at all, much les make a difficult escape. Then these censored values would not be randomly interspersed among the remaining survival times, but would constitute some of the larger values. But since these larger values would remain unobserved, you would underestimate survival probabilities beyond the 70th day.

This is known as informative censoring, and it happens more often that you might expect. Suppose someone drops out of a cancer mortality study because they are abandoning the drugs being studied in favor of laetrile treatments down in Mexico. Usually, this is a sign that the current drugs are not working well, so a censored observation here might represent a patient with a poorer prognosis. Excluding these patients would lead to an overestimate of survival probabilities.

Interpreting Kaplan-Meier plots, 1

Interpreting Kaplan-Meier plots, 2

Interpreting Kaplan-Meier plots, 3

Interpreting Kaplan-Meier plots, 4

Break #1

What you have learned
- A simple example of survival data
What’s coming next
- Overall Kaplan-Meier curve

Worcester Heart Attack Study, 1

data_dictionary: whas500.dat

description: The data represents survival times for a 500 patient subset of data from
  the Worcester Heart Attack Study. You can find more information about this data
  set in Chapter 1 of Hosmer, Lemeshow, and May.

Worcester Heart Attack Study, 2

id:
  label: a sequential code from 1 to 100
age:
  scale: ratio
  label: Age at Admission
  unit: years
gender:
  scale: binary
  value:
    Male: 0
    Female: 1

Worcester Heart Attack Study, 3

hr:
  scale: ratio
  label: Initial Heart Rate
  unit: Beats per minute
sysbp:
  scale: ratio
  label: Initial Systolic Blood Pressure
  unit: mmHg
diasbp:
  scale: ratio
  label: Initial Diastolic Blood Pressure
  unit: mmHg

Worcester Heart Attack Study, 4

bmi:
  scale: ratio
  label: Body Mass Index
  unit: kg/m^2
cvd:
  scale: binary
  label: History of Cardiovascular Disease
  value:
    'FALSE': 0
    'TRUE': 1

Worcester Heart Attack Study, 5

afb:
  scale: binary
  label: Atrial Fibrillation
  value:
    'FALSE': 0
    'TRUE': 1
sho:
  scale: binary
  label: Cardiogenic Shock
  value:
    'FALSE': 0
    'TRUE': 1

Worcester Heart Attack Study, 6

chf:
  scale: binary
  label: Congestive Heart Complications
  value:
    'FALSE': 0
    'TRUE': 1
av3:
  scale: binary
  label: Complete Heart Block
  value:
    'FALSE': 0
    'TRUE': 1

Worcester Heart Attack Study, 7

miord:
  scale: binary
  label: MI Order
  value:
    First: 0
    Recurrent: 1
mitype:
  scale: binary
  label: MI Type
  value:
    non Q-wave: 0
    Q-wave: 1

Worcester Heart Attack Study, 8

year:
  scale: ordinal
  label: Cohort Year
  value:
    yr1997: 1
    yr1999: 2
    yr2001: 3
admitdate:
  label: Admission Date
  format: mm/dd/yyyy
disdate:
  label: Hospital Discharge Date
  format: mm/dd/yyyy

Worcester Heart Attack Study, 9

fdate:
  label: Date of last Follow Up
  format: mm/dd/yyyy
los:
  scale: ratio
  label: Length of Hospital Stay
  unit: Days
dstat:
  scale: binary
  label: Discharge Status from Hospital
  value:
    Alive: 0
    Dead: 1

Worcester Heart Attack Study, 9

lenfol:
  scale: ratio
  label: Follow Up Time
  unit: days
fstat:
  scale: binary
  label: Vital Satus
  value:
    Alive: 0
    Dead: 1

Event count

# A tibble: 2 × 2
  fstat     n
  <dbl> <int>
1     0   285
2     1   215

Overall Kaplan-Meier curve

Live demo, Overall Kaplan-Meier curve

Break #2

What you have learned
- Overall Kaplan-Meier curve
What’s coming next
- The log rank test

Cox regression for gender

Call:
coxph(formula = Surv(lenfol, fstat) ~ gender, data = whas_4)

         coef exp(coef) se(coef)    z      p
gender 0.3417    1.4074   0.1526 2.24 0.0251

Likelihood ratio test=4.95  on 1 df, p=0.02606
n= 461, number of events= 176

Covariate imbalance

# A tibble: 2 × 2
  gender age_mean
   <dbl>    <dbl>
1      0     65.9
2      1     74.0

Cox regression adjusted for age

Call:
coxph(formula = Surv(lenfol, fstat) ~ gender, data = whas_4)

         coef exp(coef) se(coef)    z      p
gender 0.3417    1.4074   0.1526 2.24 0.0251

Likelihood ratio test=4.95  on 1 df, p=0.02606
n= 461, number of events= 176

Live demo, The log rank test

Break #3

What you have learned
- The log rank test
What’s coming next
- The hazard function

Life insurance example

Probabilities for ages 21 through 41

Probabilities for ages 95 through 99

Why are these probabilities not comparable?

Unequal time intervals
- Fix by computing a rate
Non-uniform probabilities over the interval
- Fix by looking at narrow interval
No adjustment for survivorship
- Fix by dividing by survival probabilty

Obviously not. There are three things you need to fix first.

The most obvious flaw is the unequal time intervals, 20 years for the first probability and 4 years for the second probability.

You can fix this by computing a rate. You get the rate by dividing the probability by the width of the time interval.

The second flaw is that the probability changes over the interval, increasing in the first case and decreasing in the second case.

You can fix this by shrinking the width of the time interval.

The third flaw is a bit more subtle. The probability of dying between the ages of 95 and 99 are probabilities computed from the perspective of a newborn child. That probability is small not because the chances of dying are small at that age, but because so many have died before their 95th birthday.

If you are in insurance sales, you do not sell policies to newborn infants. You sell to people who have survived to a certain age. No one rises from their grave on their 95th birthday and asks for an insurance policy. First, because zombies aren’t real, and second the zombie who died prior to year 95 would not be able to collect on an insurance policy that paid off for a death between 95 and 99.

You can fix this by dividing by the survivor probability.

Hazard function, definition

\[h(t)=lim_{\Delta t \rightarrow 0}\frac{P[t \le T \le t+\Delta t]/\Delta t}{P[T \ge t]}\]
\[h(t)=\frac{f(t)}{S(t)}\]
- where \(f\) is the density function, and
- \(S\) is the survival function (\(S(t)=1-F(t)\))

Hazard function, example

Hazard function on a log scale

Break #4

What you have learned
- The hazard function
What’s coming next
- The Cox regression model

Mean ages for men and women

Table of descriptive statistics showing a lower mean age for men versus women

Unadjusted and adjusted Cox regression models for gender

Unadjusted Cox regression model showing a statistically significant difference by gender

Adjusted Cox regression model shows no statistically significant difference between men and women

Live demo, The Cox regression model

Break #5

What you have learned
- The Cox regression model
What’s coming next
- Assumptions and data management

Assumptions of the log rank test

Independence
- From one patient to another
- Of censoring mechanism

There is only one assumption in the log rank test, independence. There are, however, two dimensions of independence.

First, you have to assume that the chances of an event occurring for one patient is independent of the chances of that event for another patient. There is a famous example where this was not true. Carrie Fisher, an actor and author, died on December 27, 2016. Her mother, Debbie Reynolds, herself a famous actor, died on December 28, 2016. The two events were probably related, the stress of dealing with Carrie Fisher’s death contributed to the death of Debbie Reynolds.

Often you can only assess the assumption of independence qualitatively, using your knowledge of the setting. Certain situations might make you concerned about the independence assumption, an infectious disease for example. If your data has clusters (family units, litter mates), then independence within a cluster might be questionable.

You also have to assume that when an event is censored, it is for a reason unrelated to what that event time might have been.

A patient may drop out of a study, for example, and you no longer know if or when the event occurred, except that it had to occur sometime after the date they dropped out. If a cancer patient drops out of a study because they went down to Mexico for an laetrile treatments, that might cause you to question the independence assumption. No one makes the trip to Mexico for an unapproved and unproven treatment if their current therapy was going well. The act of dropping out is informative as it indicates, at least in some cases, that death, relapse, or any other bad event is likely to happen sooner rather than later.

Assumptions of the Cox regression model

Independence
Proportional hazards assumption
Possible violations of proportional hazards
- Survival curves that cross
- One curve flattening out over time
- Curves diverge only at later times

Survival curves that cross

Survival probabilities where one group is better at early times but the other group is better at later times.

One curve flattening out over time

Survival probabilities where one curve levels off well above 0

Curves diverge only at later times

Survival probabilities that are roughly equal before day 60 but diverge afterwards

Sample size issues

Rule of 50
Rule of 15

Use ISO format for dates

XKCD cartoon satirizing the many ways that you can designate a particular date

Understand the internal storage system for dates

Screenshot of IBM SPSS website describing how dates are stored

Date management

The three dates you need
- the date of origin,
- the date of the event (if it occurred),
- the date of last contact with the patient.

The date of origin

Rehospitalization
- use date of first discharge.
Failure of a mechanical device
- use date of implant.
Divorce
- use date of marriage.
Loan default
- use date of loan contract.
Infectious disease
- use date of first exposure.

The date of the event

Define your event precisely
- All-cause mortality
- Mortality related to the health condition
- Composite endpoints (e.g., death or relapse)
  - Requires comparing the earlier of two dates.
If the event did NOT happen, leave this field blank/missing.

The date of last contact

If event did not occur
- Must be specified
- Typically last medical exam or last telephone contact
If event did occur
- Make same as event date, or
- Leave blank

Survival calculations, 1 of 2

Time = max(Date of event, Date of last contact) - Date of origin
Censoring variable = 0 if Date of event is missing, 1 if not

Survival calculations, 2 of 2

Excel spreadsheet with date calculations

Summary

What you have learned
- A simple example of survival data
- Overall Kaplan-Meier curve
- The log rank test
- The hazard function
- The Cox regression model
- Assumptions and data management